I agree with Greg. The simplest problem from one OS to another is fonts. While you can installed Microsoft licenced fonts in linux out of the box this isn't the default eg. Arial.
Even then just look at Safari for windows verusus Safari for Mac. Apple has their own implementation of the licenced MS fonts, as such the same font (eg arial) on Windows is not the same as on Mac. This can also be the case on linux if a slightly different implementation of the font is installed.
That aside, all the chrome ( toolbars, buttons, titlebar etc ) are different from one OS to another, so if you're a good developer and try really hard to word your content and fit your layout so that most people don't have to scroll just for two or three lines, then without actually viewing the page in the target OS you're really just doing half the job.
If you can get your head around it, try something like virtualbox and have a set of virtual machines, which you can run one at a time and test fully how each browser will work with your pages. A few things to note: as much as we ALL hate IE6, if your sites are going to be viewed by a company / organisation, chances are they'll still be on IE6, even worse is that there are TWO versions on IE6 which do operate slightly differently, notiable IE6 from XP ( no service packs installed ) and IE6 from XP SP2. Then you've got the default install that is Vista with IE7 ( which can look different and operate differently to IE7 on XP), and the default install in Win7 which is IE8. REALLY importantly is that it is known that some CSS on IE8 in XP is different to IE8 on Vista or Win7.
We (unfortunately) have as part of our testing 7 Win vm's to test just IE, then two for Firefox on windows ( FF 3.0 and 3.x - the latest ) plus two vms for Chrome and two vms for Safari on windows. Admittedly we promise our sites will work on all these browsers in our projects if the client chooses to at an additional cost.
Good luck